# Properties

 Label 430.2.l.a Level 430 Weight 2 Character orbit 430.l Analytic conductor 3.434 Analytic rank 0 Dimension 88 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$430 = 2 \cdot 5 \cdot 43$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 430.l (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.43356728692$$ Analytic rank: $$0$$ Dimension: $$88$$ Relative dimension: $$22$$ over $$\Q(\zeta_{12})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$88q + 4q^{6} + 12q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$88q + 4q^{6} + 12q^{7} + 16q^{13} - 88q^{16} + 4q^{17} - 16q^{21} + 12q^{23} - 8q^{25} + 12q^{28} - 36q^{30} - 40q^{31} + 12q^{33} - 40q^{35} - 40q^{36} + 16q^{38} - 56q^{41} - 56q^{43} + 24q^{46} + 72q^{47} + 24q^{50} - 16q^{52} + 20q^{53} + 24q^{55} - 8q^{56} + 20q^{57} + 8q^{60} + 72q^{61} + 36q^{62} - 40q^{66} - 16q^{67} + 4q^{68} + 24q^{71} + 60q^{73} - 48q^{76} - 48q^{77} + 40q^{78} + 28q^{81} + 20q^{86} - 64q^{87} - 8q^{90} + 48q^{91} - 12q^{92} - 108q^{93} + 4q^{95} - 4q^{96} - 48q^{97} - 96q^{98} + O(q^{100})$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1 −0.707107 + 0.707107i −0.695712 + 2.59643i 1.00000i −0.0995241 2.23385i −1.34401 2.32790i 0.420775 + 1.57036i 0.707107 + 0.707107i −3.65937 2.11274i 1.64995 + 1.50920i
7.2 −0.707107 + 0.707107i −0.547703 + 2.04406i 1.00000i 2.11862 0.715150i −1.05808 1.83265i −0.838501 3.12933i 0.707107 + 0.707107i −1.28011 0.739071i −0.992405 + 2.00378i
7.3 −0.707107 + 0.707107i −0.424244 + 1.58330i 1.00000i −1.40555 1.73909i −0.819577 1.41955i −0.632122 2.35911i 0.707107 + 0.707107i 0.271217 + 0.156587i 2.22359 + 0.235851i
7.4 −0.707107 + 0.707107i −0.388695 + 1.45063i 1.00000i −0.951195 + 2.02367i −0.750902 1.30060i 0.957274 + 3.57260i 0.707107 + 0.707107i 0.644831 + 0.372294i −0.758352 2.10355i
7.5 −0.707107 + 0.707107i −0.348214 + 1.29955i 1.00000i 1.37552 + 1.76294i −0.672697 1.16515i 0.277286 + 1.03484i 0.707107 + 0.707107i 1.03050 + 0.594958i −2.21922 0.273949i
7.6 −0.707107 + 0.707107i 0.0121292 0.0452669i 1.00000i −2.07202 + 0.840686i 0.0234319 + 0.0405852i −0.477251 1.78112i 0.707107 + 0.707107i 2.59617 + 1.49890i 0.870681 2.05959i
7.7 −0.707107 + 0.707107i 0.231077 0.862392i 1.00000i 2.23333 0.110550i 0.446407 + 0.773199i 0.738092 + 2.75460i 0.707107 + 0.707107i 1.90775 + 1.10144i −1.50103 + 1.65738i
7.8 −0.707107 + 0.707107i 0.392168 1.46359i 1.00000i −2.07474 0.833933i 0.757610 + 1.31222i 1.23476 + 4.60819i 0.707107 + 0.707107i 0.609775 + 0.352054i 2.05674 0.877385i
7.9 −0.707107 + 0.707107i 0.506805 1.89142i 1.00000i 0.195585 2.22750i 0.979072 + 1.69580i −0.522322 1.94933i 0.707107 + 0.707107i −0.722552 0.417165i 1.43678 + 1.71338i
7.10 −0.707107 + 0.707107i 0.653568 2.43915i 1.00000i 1.81492 + 1.30616i 1.26260 + 2.18688i −0.0630011 0.235123i 0.707107 + 0.707107i −2.92423 1.68830i −2.20694 + 0.359750i
7.11 −0.707107 + 0.707107i 0.867639 3.23807i 1.00000i −1.13496 + 1.92662i 1.67615 + 2.90318i 0.0566213 + 0.211314i 0.707107 + 0.707107i −7.13424 4.11896i −0.559792 2.16486i
7.12 0.707107 0.707107i −0.833140 + 3.10932i 1.00000i 1.35223 + 1.78086i 1.60950 + 2.78774i 1.22409 + 4.56838i −0.707107 0.707107i −6.37568 3.68100i 2.21543 + 0.303091i
7.13 0.707107 0.707107i −0.727882 + 2.71649i 1.00000i −2.12803 0.686647i 1.40616 + 2.43554i −0.979873 3.65693i −0.707107 0.707107i −4.25143 2.45457i −1.99028 + 1.01921i
7.14 0.707107 0.707107i −0.426111 + 1.59027i 1.00000i 1.81858 + 1.30107i 0.823184 + 1.42580i −0.688340 2.56892i −0.707107 0.707107i 0.250692 + 0.144737i 2.20592 0.365937i
7.15 0.707107 0.707107i −0.356683 + 1.33116i 1.00000i −2.17879 0.502846i 0.689058 + 1.19348i 0.982140 + 3.66540i −0.707107 0.707107i 0.953316 + 0.550397i −1.89621 + 1.18507i
7.16 0.707107 0.707107i −0.103919 + 0.387831i 1.00000i 1.79733 1.33027i 0.200756 + 0.347719i −0.783039 2.92234i −0.707107 0.707107i 2.45846 + 1.41939i 0.330259 2.21154i
7.17 0.707107 0.707107i −0.00223075 + 0.00832526i 1.00000i −1.58545 + 1.57682i 0.00430947 + 0.00746423i 0.306192 + 1.14273i −0.707107 0.707107i 2.59801 + 1.49996i −0.00610084 + 2.23606i
7.18 0.707107 0.707107i 0.0996077 0.371741i 1.00000i 2.14837 0.620072i −0.192427 0.333294i 1.10238 + 4.11412i −0.707107 0.707107i 2.46981 + 1.42594i 1.08067 1.95759i
7.19 0.707107 0.707107i 0.118826 0.443464i 1.00000i −0.834055 2.07469i −0.229554 0.397599i −0.174257 0.650336i −0.707107 0.707107i 2.41554 + 1.39461i −2.05680 0.877264i
7.20 0.707107 0.707107i 0.555341 2.07256i 1.00000i −1.79466 + 1.33387i −1.07284 1.85821i −1.21892 4.54906i −0.707107 0.707107i −1.38903 0.801955i −0.325825 + 2.21220i
See all 88 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 123.22 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
43.d odd 6 1 inner
215.l even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 430.2.l.a 88
5.c odd 4 1 inner 430.2.l.a 88
43.d odd 6 1 inner 430.2.l.a 88
215.l even 12 1 inner 430.2.l.a 88

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.2.l.a 88 1.a even 1 1 trivial
430.2.l.a 88 5.c odd 4 1 inner
430.2.l.a 88 43.d odd 6 1 inner
430.2.l.a 88 215.l even 12 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(430, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database